Luluphi udidi lwamanani okwenyani?

Los amanani okwenyani yiyo iseti yamanani aqiqayo kunye angenangqondo Ngaphezu kokukhoyo, ekunokwenzeka ukuba ufumane iintlobo ezahlukeneyo. Ezi zizalwe ngenxa yemfuneko efunyenwe phakathi kwenkulungwane ye-XV neye-XVII xa ubalo lwalungenakucaciswa ngendlela enengqiqo nechanekileyo, kuba kuqhelekile ukusetyenziswa kwamagama okanye amabinzana angenakuthenjwa, anjenge "ncinci" okanye "umda".

Nangona ama-Egypt sele esebenzisile amaqhezu, bekungekabikho kwimathematika yamaGrike apho "inani" lafundwa ngendlela yefilosofi ngakumbi, apho abalandeli bakaPythagoras bagqiba kwelokuba yonke into ebangqongileyo ngamanani; kwaye ke, ezi zisetyenziswe kwiindawo ezahlukeneyo.

Ukwahlulahlulwa kwamanani okwenyani ngokohlobo lwawo

La manani anokuhlelwa ngokweendidi ezimbini, esele sizikhankanyile ngaphambilana, oko kukuthi, amanani afanelekileyo (aqinisekileyo, amabi kunye no-zero) kunye nokungaqondakali (i-algebraic kunye ne-transcendental). Ngokuchanekileyo, kunokwenzeka ukufumana olu luhlu lulandelayo:

1. Amanani asengqiqweni

Amanani anokukwazi ukumelwa njengokwahlulwa kwamanani apheleleyo, okanye into efanayo, iqhezu eliqhelekileyo nelikhoyo ngoku apho inani elingezantsi kunye negama elingezantsi lingelilo iqanda okanye ngaphantsi kunokuba ibizwa njalo.

Ezi zinto zahlulwe zaziindidi ezahlukeneyo: iinamba ezipheleleyo (zendalo, zero kunye neenombolo ezipheleleyo) kunye namaqhezu (amaqhekeza afanelekileyo nongalunganga).

a) Amanani apheleleyo

Inani elipheleleyo lisiseti yamanani endalo, amanani apheleleyo amabi kunye no-zero, amelwe ngoonobumba "Z". Amanani apheleleyo nawo ahlala emelwe kumgca-manani, apho ii-positive okanye ezendalo zisekunene, i-zero phakathi kunye nezimbi kwi-left.

• Iyaqwalaselwa "amanani endalo”Kwabo basetyenziselwa ukubala izinto okanye ukwenza enye yeenkqubo eziqhelekileyo neziqhelekileyo zokubala.
• El zero Lixabiso elingabalulekanga oko kukuthi, liswele naliphi na inani elibalulekileyo xa lingahambi. Nangona kunjalo, indawo emi kuyo kwinani inokuyitshintsha ngokupheleleyo intsingiselo, kuba xa ikunene kwayo apho inokuphindaphinda ixabiso ngelishumi; ngelixa kwelinye icala kungekho lutshintsho.
• Los amanani apheleleyo Zisetyenziswe ngokuchaseneyo nokuhle okanye kwendalo, okt endaweni yokubala, ukusebenzisa kwabo kukuthabatha, ukutyala, ukuchitha okanye ukungaphantsi. Ukubakhankanya, kufuneka ubonise igama "thabatha" ngaphambi kwenani, umzekelo "thabatha isine".

b) Ukuqhekeka

Kwakhona ngaphakathi kwamanani okwenyani kunokwenzeka ukuba ufumane olu hlobo kwii-rationals, ezivele ngenjongo ye- Sombulula iingxaki ngokubhekisele kukwahlulwa kwamanani endalo. Inani eliyinxalenye yeqhezu libonisa nje ukwahlulahlulwa kobunye ngomnye.

Amaqhezu aphawulwa ngokuba nenani elingezantsi kunye nedinomineyitha, ezahlulwe zodwa nge-diagonal okanye kwi-bar ethe tye. Nangona kunjalo, kumanani apheleleyo singafumana "iqhezu elilula", kweli candelo iintlobo zamaqhezu esiwafumanayo zilungile kwaye azifanelekanga.

• Ezifanelekileyo ziquka ezo apho inani elingaphantsi kwedinomineyitha lingaphantsi.
• Ezingafanelekanga ziya kuba ngokuchaseneyo, oko kukuthi, idinomineyitha inkulu kunedinomineyitha.

2. Amanani angenantsingiselo

Irrationals ngala manani angenako ukubhalwa kwiqhezu, kuba iidesimali zabo ziyaqhubeka nokuziphindaphinda ngokungapheliyo. Umzekelo, akunakwenzeka ukuba ubhale iqhezu elibandakanya inani Pi, e, umlinganiselo wegolide neengcambu isikwere, ii-cubic, phakathi kwabanye.

Amanani angama-Irrational avela ngenxa yemfuneko yokuba umfundi wasePythagoras abhale ingcambu njengeqhekeza; Ukuqonda ukuba oku akunakwenzeka kwaye yayilinani esaziyo namhlanje phantsi kwegama "elingenangqondo". Nangona kunjalo, uPythagoras akavumelani nokufumanisa kwakhe, nangona kufanelekile kuye njengesikolo sakhe.

Ngapha koko, ezi zinokuhlelwa ngokweendidi ezimbini, i-algebraic kunye ne-transcendental.

• Los algebra zezo zivumela ukusombulula i-algebraic equation.
• Los ogqithileyo Zizo ezo zingenako ukumelwa linani elilinganiselweyo leengcambu (ngokungafaniyo nealgebraic) kwaye ezingalandeli mzekelo kwidesimali zazo. Phakathi kwabo sifumana inani uPi.

Ukuza kuthi ga ngoku siza nokuhlelwa kwamanani okwenyani, esinethemba lokuba kuye kwalula ukuwafunda nokuwaqonda; kuba abantu abaninzi bengabathandi bemathematics kwaye senze konke okusemandleni ethu ukunika inkcazo eneenkcukacha nelula.